A104029 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one sign Hadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-1 - 5\,x - 10\,x^{2} - 9\,x^{3} - 3\,x^{4}\right)\,P_{n-2}(x) + \left(2 + 8\,x + 10\,x^{2} + 4\,x^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 2 + 3\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	3														5
2	3	11	9													23
3	4	25	47	27												103
4	5	46	143	182	81											457
5	6	75	336	693	668	243										2021
6	7	113	676	1981	3052	2369	729									8927
7	8	161	1225	4740	10273	12620	8201	2187								39415
8	9	220	2058	10032	28483	48864	49890	27884	6561							174001
9	10	291	3264	19398	68926	154001	218356	190678	93494	19683						768101
10	11	375	4947	34986	150691	419297	770785	930634	709817	310007	59049					3390599
11	12	473	7227	59697	304414	1022385	2331621	3636659	3821574	2587083	1018595	177147				14966887
12	13	586	10241	97350	577126	2286098	6272033	12098922	16375710	15229102	9267001	3321506	531441			66067129
13	14	715	14144	152867	1038466	4765982	15376440	35587299	59420694	70999262	59205844	32716835	10761680	1594323		291634565
14	15	861	19110	232479	1788501	9376302	34955103	94907043	189702009	279028302	298327299	225447231	114093738	34676525	4782969	1287337487

Technical details

Node id: n-f2c01aaaeddbf0a7 · revision 0 · JSON
Definition source: derived:mobius_x_over_x_plus_one
Fingerprint: f6-8090db1b43d4bbd1

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:  1;
  1:  2,   3;
  2:  3,  11,    9;
  3:  4,  25,   47,    27;
  4:  5,  46,  143,   182,    81;
  5:  6,  75,  336,   693,   668,    243;
  6:  7, 113,  676,  1981,  3052,   2369,    729;
  7:  8, 161, 1225,  4740, 10273,  12620,   8201,   2187;
  8:  9, 220, 2058, 10032, 28483,  48864,  49890,  27884,  6561;
  9: 10, 291, 3264, 19398, 68926, 154001, 218356, 190678, 93494, 19683;

%F Formula

The row polynomials satisfy (1+2*x+x^2)*P(n,x) = -(1+5*x+10*x^2+9*x^3+3*x^4)*P(n-2,x) + (2+8*x+10*x^2+4*x^3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2 + 3*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(1 + 5*x + 10*x^2 + 9*x^3 + 3*x^4)*P[n - 2, x] + (2 + 8*x + 10*x^2 + 4*x^3)*P[n - 1, x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(1 + 5*x + 10*x^2 + 9*x^3 + 3*x^4)*v[i - 2] + (2 + 8*x + 10*x^2 + 4*x^3)*v[i - 1])/(1 + 2*x + x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in [-1, 0]
by Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) is the number of ternary strings of length n with k indispensable digits, with 0 <= k <= …?
    
    Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).?
    
    ↑ Liu–Wang theorem for three-term recurrences (above)

Related ?

Same fingerprint ?

A330381 → x ↦ x+1

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003